Plasmons and localized surface plasmon resonance (LSPR)

In a metal nanoparticle the light's electric field pushes the conduction electrons to one side; the positive ions pull them back, and the electron cloud oscillates. When the light's frequency matches this oscillation the particle absorbs and scatters very strongly: the localized surface plasmon resonance. It gives gold colloids their red and silver colloids their yellow colour, and its exploitation spans sensing, SERS, photothermal therapy and colour printing.

The free conduction electrons of a metal particle are displaced by the light's electric field while the positive ion lattice stays put; the surface charges this creates pull the electrons back, so they oscillate with a natural frequency set by the metal's electron density, the particle's shape and the surrounding medium. Light at that frequency drives the oscillation resonantly.

Surface plasmon resonance (SPR) usually means a propagating surface plasmon, a wave travelling along a flat metal film, excited through a prism (Kretschmann setup) and used in commercial biosensors. LSPR is the localized oscillation of a particle: no prism needed, visible as colour, with a shorter sensing range (tens of nanometres).

The metal (gold about 520 nm, silver about 390 nm, copper about 550 nm, aluminum in the UV for small spheres in water), the size (larger red-shifts), the shape (rods and shells shift to the red and near-infrared), the surrounding refractive index (higher red-shifts), and coupling to nearby particles (gaps red-shift).

For a small sphere the polarizability is α = 4πr³ (ε − ε_m)/(ε + 2ε_m), so the resonance occurs when Re ε(ω) = −2ε_m (the Fröhlich condition). With a Drude metal this gives ω_LSPR = ω_p/√(1 + 2ε_m). Larger particles need the full Mie theory, which adds the red shift and broadening from retardation.

From this site's solver, the peak shift per refractive-index unit for spheres in water is about 70 nm/RIU for 20 nm gold, 110–125 nm/RIU for 40–60 nm gold, and 115–190 nm/RIU for 20–60 nm silver. Elongated particles (nanorods) are several times more sensitive, which is why they are preferred for sensors.

As the particle grows, the field across it is no longer uniform (retardation): the restoring force weakens, so the resonance moves to longer wavelengths, and the particle radiates more, which broadens the peak. Above about 100 nm (gold) or 60 nm (silver) a second, quadrupole peak appears at shorter wavelengths.

Electrons can oscillate across the rod (transverse mode, near 520 nm) or along it (longitudinal mode). The longitudinal one red-shifts strongly with the aspect ratio: an 80 × 20 nm gold spheroid in water resonates near 800 nm.

The oscillation loses energy by electron collisions in the metal, by interband absorption (strong in gold below about 520 nm and in copper below about 590 nm), by radiation (dominant in large particles) and, in particles under about 10 nm, by electrons scattering off the surface. Silver has the least damping and the sharpest peaks.

A small region where the plasmon concentrates the light far above the incident intensity: at the poles of a sphere, the tips of a rod, and above all in the narrow gap between two particles, where |E|²/|E₀|² reaches thousands (about 1,400 for two 60 nm gold spheres 5 nm apart).

Two close gold (or silver, Ag) nanoparticles couple their plasmons: for light polarized along their axis a bonding mode red-shifts and the field in the gap grows steeply as it narrows; across the axis the shift is small and blue. When particle plasmons couple to a sharp lattice or cavity mode strongly enough to split into two hybrid modes, plasmonic systems reach the strong coupling regime, as in surface lattice resonances coupled to emitters. Such effects are exploited in sensing, SERS and nonlinear optics.

The electromagnetic enhancement of Raman scattering is about |E(ω_laser)|² |E(ω_Raman)|² / |E₀|⁴ ≈ (|E|²/|E₀|²)² for small Raman shifts. The gold dimer above gives about (1,400)² ≈ 2 × 10⁶ in its gap; chemical enhancement and the averaging over molecule positions change the measured value.

Strong, size- and shape-dependent absorption and scattering at their plasmon resonance, near-field enhancement at their surface, heating from the absorbed light, and a resonance that shifts with the surrounding refractive index. Gold and silver resonate in the visible, aluminum in the UV, copper in the green–orange; platinum and palladium have only weak, broad features.

Gold's permittivity ε(ω) combines free electrons, described by the Drude model ε = ε_∞ − ω_p²/(ω² + iγω), with interband transitions from the d band that absorb below about 520 nm. Measured data (Johnson & Christy 1972, McPeak 2015) include both and are what the calculators use; the Drude part alone misses the interband damping.

Molecules binding to a plasmonic particle raise the refractive index in its near field, which red-shifts the plasmon peak by a few nanometres; the shift is read as a colour or peak-position change. Sensitivity (nm per refractive-index unit) and line width set the detection limit; elongated particles and narrow lattice resonances do best.

Silver has the strongest, sharpest resonance but tarnishes; gold is stable and biocompatible with its resonance in the green–red and near-infrared (as rods or shells); copper is cheap but oxidizes and is more damped; aluminum covers the ultraviolet.

Yes: the absorbed light becomes heat. A 40 nm gold sphere in water under 1 mW/µm² at its resonance warms by about 23 K, which is the basis of photothermal therapy and plasmonic heating.